A review on wavelet transforms and their applications to MHD and plasma turbulence I

Transcription

1 A review on wavelet transforms and their applications to MHD and plasma turbulence I Marie Farge, Laboratoire de Météorologie Dynamique Ecole Normale Supérieure, Paris In collaboration with Kai Schneider, Aix Marseille Université Methods for Analyzing Turbulence Data Meudon, 29 May 2015

2 Choice of the appropriate representation 'It could be interesting, in communication theory, to represent an oscillating signal by a superposition of elementary wavelets, each of them having both a frequency and a time localization quite well defined. The useful information is indeed often carried by both the emitted frequencies and by the time structure of the signal (music is a characteristic example of that). The representation of a signal as function of time cannot exhibit the frequency content, while in contrast its Fourier analysis hides the time of emission and the duration of each elements of the signal. An adequate representation should combine the advantages of these two complementary descriptions, while providing a discrete character appropriate to the theory of communication.' Roger Balian! Un principe d'incertitude! fort en théorie du signal! CRAS, 292, II (1981)!

3 Representation for music Guido d Arezzo Micrologos tones: ut re mi fa sol la si sa re ga ma pa da ni 12 half tones:

4 Integral transforms Δx Δk = A Gridpoint Fourier (1807) A A

5 Optimal phase space tiling Gabor (1946) Wavelets (1984) Space-wavenumber representation Δx Δk = information atom Space-scale representation

6 Choice of the analyzing wavelet Admissibility condition Jean Morlet Alex Grossmann Analyzing wavelet family generated by translation (b) and dilation (a) Grossmann and Morlet, Decomposition of Hardy functions into square integrable wavelets of constant shape, SIAM J. math. Anal., 15(4), , 1984

7 Continuous wavelet transform (CWT) Analysis Synthesis Parseval s identity

8 Wavelet representation Physical space Δx j = 7 Δx Δk > C ψ ji j = 6 j = 5 j = 4 j = 3 i Spectral space Δk ˆ ψ ji Farge, Wavelet transforms and their applications to turbulence Ann. Rev. Fluid Mech., 92, 1992 Farge and Schneider, Wavelets: application to turbulence, Encyclopedia of Mathematical Physics, Springer, , 2006

9 2D continuous wavelet transform Romain Murenzi Jean-Pierre Antoine 2D Morlet mother wavelet The wavelet family is generated by translating, dilating and rotating the 2D mother wavelet

10 Analyzing wavelet Field to analyze Modulus of the wavelet coefficients Small scale Large scale Logarithm

11 Reproducing kernel of the CWT The CWT of a Gaussian white noise reveals its reproducing kernel Modulus Phase The is the correlation between the wavelets which corresponds to the redundancy between the coefficients

12 Wavelet frame We can then select a finite number of wavelets restricted to a discrete grid optimally chosen such that the wavelet family associated to this grid constitutes a quasi-orthogonal basis a wavelet frame For example for Marr wavelet we need a 0 =2 1/2 b 0 =1/2

13 Orthogonal wavelet transform Wavelet analysis : with Wavelet synthesis : A signal sampled on N points is wavelet analyzed and synthetized in CN operations if one uses compactly-supported wavelets computed from a quadratic mirror filter of length M.

14 Examples of orthogonal wavelets

15 2D orthogonal wavelets Scaling function Wavelet Coarse approximation Horizontal details Vertical details Diagonal details

16 3D orthogonal wavelets A 3D vector field v(x) sampled on 3D wavelet equidistant grid points orthogonal wavelet series N j = j, wavelet coefficients at a scale indexed by j fast algorithm with linear complexity no redundancy between the coefficients We use Coifman 12 wavelet compactly supported with four vanishing moments.

17 Orthogonal wavelet representation Wavelets ψ ji ~ f Wavelet coefficients ji = ψ ji f scale j N = 512 = 2 9 position i Mallat, A wavelet tour of signal processing, 3rd edition, Academic Press, 2008

18 Academic example Function sampled on N = 512 grid-points N = 512 wavelet coefficients

19 Linear approximation Function reconstructed from 64 wavelet coefficients 64 wavelet coefficients such that j 6

20 Nonlinear approximation Function reconstructed from 64 wavelet coefficients 64 wavelet coefficients such that

21 Wavelet analysis Grossmann Meyer Daubechies Mallat

22 Continuous / orthogonal wavelets Analyzing functions are translates and dilates of an oscillating function (of zero mean) Well localized in both space and wavenumber Continuous wavelets Orthogonal wavelets Translates and dilates vary continuously Redundant representation Coefficients are easy to read Unfold in both space and scale Good for analysis Translates and dilates are on a discrete dyadic grid Orthogonal basis Coefficients not easy to read sampled on a dyadic grid For filtering and compression

23 How to extract coherent structures? Since there is not yet a universal definition of coherent structures which emerge out of turbulent fluctuations due to the nonlinear interactions, we adopt an apophetic method : instead of defining what they are, we define what they are not. For this we propose the minimal statement : Coherent structures are not noise Extracting coherent structures becomes a denoising problem, not requiring any hypotheses on the structures themselves but only on the noise to be eliminated. Choosing the simplest hypothesis as a first guess, we suppose we want to eliminate an additive Gaussian white noise, and for this we use a nonlinear wavelet filtering. Farge, Schneider et al., Phys. Fluids, 15 (10), 2003 Azzalini, Farge, Schneider, ACHA, 18 (2), 2005

24 Denoising using wavelets Gaussian white noise is by definition equidistributed among all modes and the amplitude of the coefficients is given by its r.m.s., whatever the functional basis one considers. Therefore the coefficients of a noisy signal whose amplitudes are larger than the r.m.s. of the noise belong to the denoised signal. This procedure corresponds to nonlinear filtering. The advantage of performing such a nonlinear filtering using the wavelet representation is that the wavelet coefficients preserve the space locality, since wavelets are functions localized in both physical and spectral space. Since we do not know a priori the r.m.s. of the noise, we have proposed an iterative procedure which takes as first guess the r.m.s. of the noisy signal. Azzalini, M. F., Schneider, 2005 Appl. Comput. Harmonic Analysis, 18 (2)

25 Wavelet denoising algorithm Apophatic method : - no hypothesis on the structures, - only hypothesis on the noise, - simplest hypothesis as our first choice. f Hypothesis on the noise : f n = f d + n n Gaussian white noise, <f n2 > variance of the noisy signal, N number of coefficients of f n. Wavelet decomposition : f ji =< f ψ ji > Estimation of the threshold : ε n = 2 < f n 2 > ln(n) Wavelet reconstruction : f d = ji: f ~ ji >ε n f ~ ji ψ ji j scale, i position Donoho, Johnstone, Biometrika, 81, 1994 Azzalini, M. F., Schneider, ACHA, 18 (2), 2005 f n f d

26 1. Application to plasma turbulence in tokamaks Tore-Supra, Cadarache (France) ETE, INPE (Brazil) JET, Culham (Europe) ITER (World)

27 Extraction of coherent structures SOL Ion density fluctuations measured by a fast reciprocating Langmuir probe in the SOL of the tokamak Tore Supra (Pascal Devynck, Tore-Supra, CEA-Cadarache) + Coherent Farge, Schneider & Devynck Phys. Plasmas,13, , 2006 Incoherent

28 PDF of the density fluctuations -5/2 Total fluctuations = coherent + incoherent fluctuations Farge, Schneider & Devynck, Phys. Plasmas,13, 2006

29 Correlation and intermittency Scalogram (stabilized periodogram) Flatness versus scale (from wavelet coefficients) Coherent ones are correlated k 0 k -5/3 Coherent fluctuations are intermittent Incoherent fluctuations are decorrelated 3 Incoherent ones are non-intermittent Total fluctuations = coherent + incoherent fluctuations Farge, Schneider & Devynck, Phys. Plasmas,13, 2006

30 2. Applications in 2D fluid turbulent flows DNS N= % of coefficients 99.8 % of kinetic energy 93.6 % of enstrophy Coherent flow 99.8 % of coefficients 0.2 % of kinetic energy 6.4 % of enstrophy Total flow Incoherent flow + +ωmin +ωmax

31 1D cut of the vorticity field DNS N=512 2 Total Coherent 0.2 % N 99.8 % E 93.6 % Z ω t =ω c +ω i Z t =Z c +Z i Incoherent 99.8 % N 0.2 % E 6.4 % Z

32 PDF of vorticity DNS N=512 2 Total log p(ω) Coherent 0.2 % N 99.8 % E 93.6 % Z Incoherent 99.8 % N 0.2 % E 6.4 % Z ω min ω max

33 Enstrophy spectrum DNS N=512 2 log Z(k) Coherent 0.2 % N 99.8 % E 93.6 % Z Incoherent k -1 scaling, i.e. enstrophy equipartition since E=k -2 Z Coherent Total k -5 scaling, i.e. long-range correlation Incoherent 99.8 % N 0.2 % E 6.4 % Z log k

34 A posteriori proof of coherence DNS N=512 2 Coherent structures are locally (in space and time) steady solutions of Euler equation, thus, for 2D flows : Arnold, 1965, Joyce & Montgomery, 1973 Robert & Sommeria, 1991 ω = sinh(ψ) Total Coherent ω ω ω Incoherent ψ ψ

35 3. Application to 3D fluid turbulence Dissipation rate α Kaneda et al., 2003 Phys. Fluids, 12, Transition Fully-developed turbulence R λ 1200 Dissipation rate is independent of viscosity turbulent dissipation How turbulent dissipation differs from viscous dissipation?

36 3D homogeneous isotropic turbulence Resolution N= π L, integral scale Computed in 2002 on ES1 14 Tflops 10 Tbytes Kaneda, Ishihara et al., 2003, Phys. Fluids, 12, L

37 Zoom (sub-cube ) Resolution N= L, Integral scale L λ Kaneda, Ishihara et al., 2003, Phys. Fluids, 12, λ, Taylor microscale

38 Zoom (sub-cube ) Resolution N= L L, integral scale λ, Taylor macroscale Kaneda, Ishihara et al., 2003, Phys. Fluids, 12, λ

39 Zoom (sub-cube ) Resolution N= λ, Taylor macroscale η, Kolmogorov dissipative scale λ η Kaneda, Ishihara et al., 2003, Phys. Fluids, 12, 21-24

40 Zoom (sub-cube 1283 ) DNS N=20483 Kaneda, Ishihara et al., 2003, Phys. Fluids, 12, 21-24

41 Zoom (sub-cube 64 3 ) DNS N= Kaneda, Ishihara et al., 2003, Phys. Fluids, 12, 21-24

42 Extraction of coherent structures in 3D flows DNS N= ω =5σ with σ=(2ζ) 1/2 Coherent vorticity 2.6 % N coefficients 80% enstrophy 99% energy Incoherent vorticity 97.4 % N coefficients 20 % enstrophy 1% energy Total vorticity + ω =5σ Okamoto, Yoshimatsu, Schneider, Farge, Kaneda, 2007, Phys. Fluids, 19, 1159 ω =5/3σ

43 Energy spectrum DNS N= k -5/3 log E(k) k +2 Okamoto, Yoshimatsu, Farge, Schneider, Kaneda, 2007 Phys. Fluids, 19(11) 2.6 % N coefficients 80% enstrophy 99% energy Multiscale Coherent k -5/3 scaling, i.e. long-range correlation Multiscale Incoherent k +2 scaling, i.e. energy equipartition log k

44 PDF of velocity DNS N= log p(v) Okamoto, Yoshimatsu, Farge, Schneider, Kaneda, 2007 Phys. Fluids, 19(11) 2.6 % N coefficients 99% energy v The total and coherent flows have the same extrema. The incoherent flow has a Gaussian PDF, therefore its effect should be easy to model

45 Nonlinear transfers and energy fluxes ccc coherent flux = total flux L Inertial range iic, iii incoherent flux = 0 η cci ttt icc, iic

46 You can download movies from : Results You can download papers from : Publications You can download codes from : Codes

47 Review papers on wavelets Marie Farge, 1992 Wavelet Transforms and Their Applications to Turbulence Ann. Rev. Fluid Mech., 24, Marie Farge, Nicholas Kevlahan, Valerie Perrier and Eric Goirand, 1996 Wavelets and Turbulence IEEE Proceedings, 84, 4, 1996, Marie Farge, Nicholas Kevlahan, Valérie Perrier and Kai Schneider, 1999 Turbulence Analysis, Modelling and Computing using Wavelets Wavelets in Physics, ed. J. van den Berg, Cambridge University Press, Marie Farge and Kai Schneider, 2002 Analyzing and computing turbulent flows using wavelets Summer Course, Les Houches LXXIV, New trends in turbulence, Springer Kai Schneider and Marie Farge, 2006 Wavelets: Mathematical Theory Encyclopedia of Mathematical Physics, Elsevier, Marie Farge and Kai Schneider, 2006 Wavelets: Application to Turbulence Encyclopedia of Mathematical Physics, Elsevier,

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